where Greek indices \(\mu ,\nu ,\cdots \)  represent
spacetime indices and \(\nabla _\mu \)  denotes the covariant derivative with respect to
the metric \(g_{\mu \nu }\) ,
with the scalar field \(\phi \)  unchanged.
Note that in the present paper, for simplicity, a “disformal transformation” denotes
a transformation of the form (REF ) with \(F_0\ne 1\)  and \(F_1\ne 0\)  in general (and also its generalization discussed below), which also contains the purely conformal transformation with \(F_1=0\)  as a special case.
As we shall see in detail in §REF , the transformation (REF ) is invertible (i.e., the new-frame metric \(g_{\mu \nu }\)  can be uniquely expressed in terms of \(\phi \)  and \(\bar{g}_{\mu \nu }\)  without referring to any particular configuration) if and only if \(F_0(F_0+XF_1)(F_0-XF_{0X}-X^2F_{1X})\ne 0\) , where \(F_{iX}\partial F_i/\partial X\)  (\(i=0,1\) ).
Interestingly, it was shown in [1]} that the disformal transformation (REF ) in general maps the Horndeski class to its exterior, which is now known as the quadratic/cubic DHOST class [2]}, [3]}, [4]}.However, in two spacetime dimensions, the Horndeski class is closed under the disformal transformation of the form (REF ) [5]}.
Also, the quadratic/cubic DHOST class itself is closed under the disformal transformation [4]}.
Applications of the disformal transformation have been extensively explored in various contexts; inflation [7]}, [8]}, [9]}, [10]}, [11]},
cosmic microwave background [12]}, [13]}, [14]},
dark matter and dark energy [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]},
cosmological perturbations [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]},
solar system tests and screening mechanisms [30]}, [16]}, [32]}, [33]}, [34]}, [35]},
relativistic stars [36]}, [37]}, [38]}, [39]},
and black holes [40]}, [41]}, [42]}, [43]}, [44]}, [45]}, [46]}, [47]}, [48]}, [49]}, [50]}, [51]}, [52]}, [53]}.
